ANSWERS 287

ANSWERS

1.3 EXERCISE

1. (b,b), (c,c), (a,c)

2. [-5,5]

3. 24 4 –1x x+

4. ( )1 3

2

xf x− +=

5. ( ) ( ) ( ){ }1 , , , , , ,( , )f b a d b a c c d−

6. ( )( ) 4 3 2– 6 10 – 3f f x x x x x= +

7. 2, –1 = =

8. (i) represents function which is surjective but not injective

(ii) does not represent function.

9. ( ) ( ) ( ){ }2,5 , 5,2 , 1,5fog =

12. (i) f is not function (ii) g is function (iii) h is function (iv) k is not function

14.1

,13

17. Domain of R = {1,2,3,4, ..... 20} and

Range of R = {1,3,5,7,9, ..... 39}. R is neither reflective, nor symmetric and nortransitive.

21. (i) f is one-one but not onto , (ii) g is neither one-one nor onto (iii) h is bijective,(iv) k is neither one-one nor onto.

22. (i) transitive (ii) symmetric (iii) reflexive, symmetric and transitive (iv) transitive.

23. ( ) ( ) ( ) ( ) ( ){ }2,5 1,4 , 2,5 , 3,6 , 4,7 (5,8),(6,9) =

288 MATHEMATICS

25. (i) ( )( ) 24 – 6 1fog x x x= +

(ii) ( )( ) 22 6 –1gof x x x= +

(iii) ( )( ) 4 3 26 14 15 5fof x x x x x= + + + +

(iv) ( )( ) 4 – 9gog x x=

26. (ii) & (iv)

27. (i) 28. C 29. B 30. D

31. B 32. B 33. A 34. C

35. C 36. B 37. D 38. A

39. B 40. B 41. A 42. A

43. C 44. B 45. D 46. A

47. B 48. ( ) ( ){ }R = 3,8 , 6,6 , (9,4), (12,2)

49. ( ) ( ) ( ){ }R 1,1 , 1,2 , 2,1 ,(2,2),(2,3), (3,2), (3,3), (3,4), (4,3), (4,4), (5,5)=

50. ( ) ( ) ( ){ } ( ) ( ) ( ){ }1,3 , 3,1 , 4,3 and 2,5 , 5,2 , 1,5gof fog= =

51. ( )( )23 1

xfofof x

x=

+52. ( ) ( )

1

3–1 7 4 –f x x= +

53. False 54. False 55. False 56. False

57. True 58. False 59. False 60. True

61. False 62. False

2.3 EXERCISE

1. 0 2. – 1 4.–π12

5. –π3

7. 0, –1 8.14

1511.

–3 3,

4 4

ANSWERS 289

13. –1 4tan –3

x 17.4

19.

1

1

–1

n

n

a a

a a+

20. C 21. D 22. B 23. D

24. A 25. A 26. B 27. C

28. A 29. B 30. A 31. D

32. D 33. B 34. A 35. C

36. A 37. A

38.2π3

39.2π5

40. 3 41. φ

42.π3

43.2π3

44. 0 45. 1

46. –2π,2π 47. xy > – 1 48. –1π – cot x

49. False 50. False 51. True 52. True

53. True 54. False 55. True

3.3 EXERCISE

1. 28 × 1, 1 × 28, 4 × 7, 7 × 4, 14 × 2, 2 × 14. If matrix has 13 elements then its orderwill be either 13 × 1 or 1 × 13.

2. (i) 3×3, (ii) 9, (iii) 223 31 12– , 0, 1a x y a a= = =

3. (i)

1 9

2 20 2

(ii)1 4

–1 2

4.2 2

3 3

sin sin 2

sin sin 2

sin sin 2

x x

x x

x x

e x e x

e x e x

e x e x

5. a = 2, b = 2 6. Not possible

7. (i)5 2 –2

X Y12 0 1

+ =

(ii)

0 –1 12X 3Y

–11 – 10 –18

− =

290 MATHEMATICS

(iii) 5 – 2 2

Z–12 0 –1−

=

8. x = 4 10. – 2, – 14

11.–1 –2 –3–1

A1 57

=

12.

1 1A =

1 0

13. A = [– 1 2 1]

15.

9 6 1212 9

AB= BA 7 8 1612 15

4 5 10

=

18. x = 1, y = 2

19.– 2 0 2 1

X ,Y–1 – 3 2 2

= =

20. ,

2 2 2

k k k

k k k

etc.

where k is a real number

24. A = [– 4] 30. True when AB = BA

37. (i)7 -31

5 122

(ii) not possible

38. x = 2, y = 4 or x = 4, y = 2, z = – 6, w = 4

39.–24 –10–28 –38

40.3 187 –195

A–156 148

=

41. a = 2, b = 4, c = 1, d = 3 42.1 –2 –53 4 0

43.18 8

16 18

44.True for all real values of α

45. a = – 2, b = 0, c = – 3

ANSWERS 291

50.1 1 1

, ,2 6 3

x y z= ± = ± = ±

51. (i)

7 9 10

12 15 17

1 1 –1

− − − −

(ii) inverse does not exist (iii)

3 1 1

15 6 5

5 2 2

− − − −

52.

5 32 2 0 1

2 23 1

2 1 1 02 2

5 3 3 12 0

2 2 2 2

− − + − −

53. A 54. D 55. B 56. D

57. D 58. D 59. A 60. B

61. C 62. D 63. A 64. A

65. D 66. D 67. A 68. Null matrix

69. Skew symmetric matrix 70. – 1 71. 0

72. Rectangular matrix 73. Distributive

74. Symmetrix matrix 75. Symmetrix matrix

76. ( )(i)B A (ii) A (iii) A –Bk k′ ′ ′ ′ ′ 77. Skew Symmetric matrix

78. (i) Skew symmetric matrix

(ii) neither symmetric nor skew symmetric matrix

79. Symmetric matrix 80. AB = BA 81. does not exist

82. False 83. False 84. False 85. True

86. True 87. False 88. False 89. True

90. False 91. False 92. False 93. False

94. True 95. False 96. True 97. False

98. True 99. False 100. True 101. True

292 MATHEMATICS

4.3 EXERCISE

1. x3 – x2 + 2 2. a2 (a + x + y + z) 3. 2x3y3z3

4. 3 (x + y + z) (xy + yz + zx) 5. 16 (3x + 4) 6. (a + b + c)3

12. ( )θ π or π + –16

nn n

= 13. x = 0, – 12 18. x = 0, y = – 5, z = – 3

19. x = 1, y = 1, z = 1 20. x = 2, y = – 1, z = 4

24. C 25. C 26. B 27. D

28. C 29. A 30. A 31. A

32. C 33. D 34. D 35. D

36. B 37. C 38. 27 A 39.1

A

40. Zero 41.1

242. (A–1)2 43. 9

44. Value of the determinant 45. x = 2 y = 7

46. (y – z) (z – x) (y – x + xyz) 47. Zero 48. True

49. False 50. False 51. True 52. True

53. True 54. False 55. True 56. True

57. True 58. True

5.3 EXERCISE

1. Continuous at x =1 2. Discontinuous 3. Discontinuous 4. Continuous

5. Discontinuous 6. Continuous 7. Continuous 8. Discontinuous

9. Continuous 10. Continuous 11.7

2k = 12.

1

2k =

13. k = –1 14. 1k = ± 16. a = 1, b = –1

17. Discontinuous at x = – 2 and–52

x= 18. Discontinuous at x = 1,1

2 and 2

20. Not differentiable at x = 2 21. Differentiable at x = 0

22. Not differentiable at x = 2 25.2cos– (log 2) sin 2 2 xx⋅ ⋅

ANSWERS 293

26. 8

8 8log8

x

xx

− 27. 2

1

x a+ 28. ( ) ( )5 5

5

log log logx x x

29.cos sin 2–2 2

x x

x x30. ( ) ( ) ( )–1 2 22 sin cosnn ax b ax bx c ax bx c+ + + + +

31. ( ) ( )2–1sin tan 1 sec 1

2 1x x

x+ +

+

32. ( ) ( )2 22 cos 2 sin 2 sin 2x x x x x+ + 33. ( )–1

2 1x x +

34. ( )2

cos cossin – sin .logsinsin

x xx x x

x

35. ( )sin cos – tan cotmx nx x n x m x+

36. ( )( ) ( )2 3 21 2 3 9 34 29x x x x x + + + + +

37. – 1 38.1

239.

1

240. – 1

41. 2

–3

1 – x42. 2 2

3a

a x+ 43. 4

–

1 –

x

x44.

2

2

1

–1t

t

+

45.3 2

2θ3 2

-θ +θ +θ+1θ +θ +θ–1

e−

46. cot θ 47. 1

48. t 51.1

3− 52. 2

tan –sin

x x

x53.

1

2

54.( )

( )2 3

2 2

2 – cos –cos –

xy y xy y

xy xy x y+ 55.( ) ( )

( ) ( )sec tan

sec tan –y x y x y

x y x y x

− + ++ +

56.–x

y 57.3 2

2 3

– 4 – 44 4 –y x xy

yx y x+ 64. 3–2sin cosy y

70. Not applicable since f is not differentiable at x = 1

294 MATHEMATICS

71. ( ), – 2 72. (2, –4) 77.7 1

,2 4

78.

3, 0

2

79. 3, 5p q= = 82. xtanx2

2

tansec log

2 1

x xx x

x x

+ + + 83. D

84. C 85. B 86. A 87. A

88. A 89. C 90. B 91. B

92. A 93. A 94. B 95. A

96. B 97. –1x x+ 98.2

3x99.

–12

100.3 1

2

+

101. – 1 102. False 103. True

104. True 105. True 106. False

6.3 EXERCISE

3. 8 m/s 4. ( )2 – 2 v unit/sec. 5.πθ3

= 6. 31.92

7. 0.018πcm3 8.2

23

m/s towards light, –1 m/s

9. 2000 litres/s, 3000 litre/s 11. 2x3 – 3x + 1

12. k2 = 8 14. (4, 4) 15.1 4 2

tan7

−

17. 3 8x y+ = ±

18. (3, 2), (–1, 2) 23. (1, – 16), max. slope = 1226. x = 1 is the point of local maxima; local maximum = 0

x = 3 is the point of local minima; local minimum = – 28x = 0 is the point of inflection.

27. Rs 100 30. 6cm, 12 cm, 3864 cm

ANSWERS 295

31. 1:1 33. Rs 1920 34.32 2π

13 27

x +

35. C 36. B 37. A 38. C

39. D 40. A 41. A 42. D

43. B 44. B 45. C 46. B

47. D 48. A 49. B 50. C

51. A 52. C 53. B 54. C

55. B 56. A 57. B 58. B

59. C 60. (3, 34) 61. x + y = 0 62. ( )– , –1∞

63. (1, ∞ ) 64. 2 ab

7.3 EXERCISE

3.2

– 3log 12

xx x c+ + + 4.

3

3

xc+ 5. log sinx x c+ +

6. tan C2

x + 7.5 3tan tan

5 3

x xc+ + 8. x + c

9. –2cos 2sin2 2

x xc+ + 10. 2 – – log 1

3 2

x x xx x c

+ + +

11.

2–1

2cos 1

x xa c

a a

− + − + 12.

33/ 4 44 – log 1

3x x c

+ +

13.

3

2

2

–1 11

3c

x

+ + 14. –11 3

sin3 4

xc+

15.11 4 –3

sin32

tc− +

16. 2 23 9 – log 9x x x c+ + + +

296 MATHEMATICS

17. 2 2–1 5 – 2 2log –1 5– 22

xx x x x x c+ + + + +

18. { }2 21 log –1 – log 14

x x c+ + 19.11 1 1log – tan

4 1 2

xx c

x− + + −

20.2

2 1– –2 – sin2 2

x a a x aax x c

a− + +

21.1

2

2

sinlog 1

1–

x xx

x

−

+ −

22. –1

sin 2 sin2

x x c + + 23. tan x – cot x – 3x + c

24.3

13

2sin

3

xc

a− + 25. 2 sin x + x + c

26.1 21

sec ( )2

x c− + 27.26

3

28. 2 –1e 29. 1tan –4

e− 30. 2

log

–1m

m 31. π

32. 2 –1 33.3

34. 12 2

tan2 3

−

35.–11 – 2 3

log tan7 2 7 3

x xc

x+ +

+

36.1 1

2 2

1tan tan

–x x

a b ca ba b

− − − +

37. π

38.( ) ( )

1 1

6 3

– 3log

–1 2

xc

x x+

+39.

–1tan x

xe c+

40.–1 –1tan tan

x x x xa c

a a a a

− + +

41.

3

2

ANSWERS 297

42. [ ] [ ]3 33

sin 3 cos3 sin 3cos24 40

x xe ex x x x c

− −

− + − +

43.11 tan –1 1 tan – 2 tan 1

tan log2 2 tan 2 2 tan 2 tan 1

x x x

x x x− ++ + +

+ c

44.2 2

3 3

π4

a b

a b

+

45.3

log38

46.2π 1

log2 2

47.π 1

log4 2

48. A 49. C 50. A 51. C

52. D 53. C 54. D 55. D

56. D 57. A 58. D 59. e –1

60.4

xec

x+

+61.

1

262.

–1–1 2costan

2 3 3

xc

+ 63. 0

8.3 EXERCISE

1.1

sq.units2

2.24

3p sq. units 3. 10 sq.units 4.

16sq.units

3

5.27

2sq.units 6.

9

2sq. units 7.

32

3 sq. units 8. 2π sq.units

9.4

sq.units3 10. 96 sq.units 11.

16sq.units

312.

2π4

a sq. units

13.1

6 sq. units 14.

9

2sq. units 15. 9 sq.units 16.

82 π sq.units

3 −

17. 4 sq.units 18.15

2sq. units 19. ( ) 24

3 2π3

a+ sq. units

20. 6 sq.units 21.15

2sq. units 22. 8 sq.units 23. 15 sq.units

24. C 25. D 26. A 27. B

298 MATHEMATICS

28. A 29. A 30. D 31. A

32. B 33. A 34. C

9.3 EXERCISE

1. –2 – 2x y k− = 2.2

20

d y

dx= 3.

6 9

2

e +

4. ( )2 1 –1–1 log2 1

xy x k

x

= + + 5.

2–. x xy c e=

6. ( ) mx axa m y e ce−+ = + 7. (x – c) ex+y + 1 = 0

8.2–

2

x

y kxe= 9.2

tan2

xy x

= + 10. ( )2x y y c= + 11.

1

3

13.2

22

(1 – ) – 2 0d y dyx x

dxdx− = 14. ( )2 2– – 2 0dy

x y xydx

=

15. ( )3

2

4

3 1

xy

x=

+ 16.1tan log

yx c

x− = +

17.1 1tan 2 tan2 y yxe e c

− −

= + 18.–1tan log

xy c

y

+ =

19. –x yx y k e+ = 20. ( )32 23 yx y e ++ = 21.– cos2 3

sin2 2

xy x = +

22. ( )2 – 0xy y x y yy′′ ′ ′+ = 23. ( ) ( )2–1 21tan log 1

2x y c+ + =

24. ( ) ( )–1 – 2 0dyx y

dx+ = 25.

–22

2sin 2cos log– cos –3 9

x x x x xy x cx

x x= + + + +

ANSWERS 299

26. ( ) –sin cos sin yx y y y ce+ = + 27. log 1 tan2

x yx c

+ + = +

28.33sin 2 2cos2–

13xx x

y ce+ = +

29. ( )2 22 – 3x y x=

30. ( )( )–1 1 2 0y x x+ + = 31. k ( )2 1 – 1 –xe x y x y+ = +

32. 1xy = 33. logx

cxy

=

34. D 35. C

36. A 37. C 38. B 39. C

40. C 41. D 42. A 43. C

44. D 45. B 46. B 47. C

48. C 49. D 50. A 51. A

52. B 53. B 54. B 55. B

56. C 57. B 58. A 59. A

60. C 61. C 62. D 63. C

64. C 65. A 66. D 67. D

68. C 69. C 70. A 71. A

72. A 73. C 74. B 75. A

76. (i) not defined (ii) not defined (iii) 3

(iv) Qdy

pydx

+ = (v)1 1

1Qp dy p dy

xe e dy c ∫ ∫= × + ∫

(vi)2

2

4

xy cx−= + (vii) ( )2 33 1 4y x x c+ = +

(viii) xy = Ae–y (ix)– sin cos–

2 2x x x

y ce= +

(x) x = c sec y (xi)xe

x

77. (i) True (ii) True (iii) True (iv) True

(v) False (vi) False (vii) True (viii) True

(ix) True (x) True (xi) True

300 MATHEMATICS

10.3 EXERCISE

1. ( )12 2

3i j k+ + 2. (i) ( )1 2 – 2

3i j k+ (ii)

( )16

37j k+

3. ( )1 –2 3 – 67

i j k+ 4.3 –

2

b ac =

5. k = –2 6. ( )2 i j k± + +

7. 2 3 –6, , ;4 ,6 , –127 7 7

i j k 8. 2 4 4i j k− + + 9.–1 1

cos156

10. Area of the parallelograms formed by taking any two sides represented by ,a b and

c as adjacent are equal

11.2

712. 21 13.

274

2

16. a b b c c an

a b b c c a

× + × + ×=× + × + ×

17.

62

2

18. ( )15 2 2

3i j k+ +

19. C 20. D 21. C 22. B

23. D 24. A 25. D 26. D

27. D 28. A 29. C 30. A

31. C 32. C 33. B

34. If a and b are equal vectors

35. 0 36.4

37. ] [ 1–1,1 –

2k k∈ ≠ 38.

2 2a b

39. 3 40. a

41. True 42. True

43. True 44. False 45. False

11.3 EXERCISE

1. ˆˆ ˆ5 +5 2 +5i j k 2. ˆ ˆˆ ˆ ˆ ˆ( –1) + ( +2) + ( – 3) = (3 – 2 + 6 )x i y j z k j j k

3. (–1, – 1, – 1)

ANSWERS 301

4.–1 19

cos21

7. x + y + 2z = 19 8. x + y + z = 9

9. 3x – 2y + 6z – 27 = 0 10. 21x + 9y – 3z – 51 = 0

11. and1 2 –1 –1 1 –2x y z x y z= = = = 12. 60°

14. ax + by + cz = a2 + b2 + c2 14. (1, 1)

15. 15° or 75° 16. (2, 6, –2) 3 5

17. 7 18. 6

19. ˆ ˆˆ ˆ ˆ ˆ( – 3) + y + (z –1) = (–2 + +3 )x j j k i j k

20. 18x + 17y + 4z = 49 21. 14 22. 51x + 15y – 50z + 173 = 0

24. 4x +2y – 4z – 6 = 0 and –2x + 4y + 4z – 6 = 0

26. ˆ ˆˆ ˆ ˆ ˆ3 +8 +3 , – 3 – 7 6i j k i j k+ 29. D 30. D

31. A 32. D 33. D 34. A

35. D 36. C 37. 12 3 4

x y z+ + =

38.2 2 –1

, ,3 3 3

39. ˆ ˆˆ ˆ ˆ ˆ( – 5) ( 4) ( – 6) (3 + 7 + 2 )x i y j z k i j k+ + + =

40. ˆ ˆˆ ˆ ˆ ˆ( – 3) ( – 4) ( 7) (–2 – 5 + 13 )x i y j z k i j k+ + + = 41. x + y – z = 2

42. True 43. True 44. False 45. False

46. True 47. True 48. False 49. True

12.3 EXERCISE

1. 42 2. 4 3. 47 4. – 305. 196 6. 43 7. 21 8. 47

9. Minimum value = 3 10. Maximum = 9, minimum = 31

7

302 MATHEMATICS

11. Maximise Z = 50 60 ,x y+ subject to:

2x + y ≤ 20, x + 2y ≤ 12, x + 3y ≤ 15, x ≥ 0, y ≥ 0

12. Minimise Z 400 200x y= + , subject to:

5 2 30

2 15

, 0, 0

x y

x y

x y x y

+ ≥+ ≤

≤ ≥ ≥

13. Maximise Z = 100 170x y+ subject to :

3 2 3600, 4 1800, 0, 0x y x y x y+ ≤ + ≤ ≥ ≥14. Maximise Z = 200 120x y+ subject to :

300, 3 600, 100, 0, 0x y x y y x x y+ ≤ + ≤ ≤ + ≥ ≥

15. Maximise Z = ,x y+ subject to

2x + 3y ≤ 120, 8x + 5y ≤ 400, x ≥ 0, y ≥ 0

16. Type A : 6, Type B : 3; Maximum profit = Rs. 480

17. 2571.43 18. 138600

19. 150 sweaters of each type and maximum profit = Rs 48,000

20.2

54 km.7

21.10

311

22. Model X : 25, Model Y : 30 and maximum profit = Rs 40,000

23. Tablet X : 1, Tablet Y : 6 24.Factory I : 80 days, Factory II : 60 days

25. Maximum : 12, Minimum does not exist

26. B 27. B 28. A 29. D

30. C 31. D 32. D 33. A

34. B 35. Linear constraints36. Linear 37. Unbounded

38. Maximum 39. Bounded 40. Intersection 41. Convex

42. True 43. False 44. False 45. True

ANSWERS 303

13.3 EXERCISE

1. Independent 2. not independent 3. 1.1 4.25

56

5.1 5 7

P(E) = , P(F) : ,P(G) = ,12 18 36

no pair is independent

7. (i)3

4, (ii)

1

2, (iii)

1

4, (iv)

5

88.

3 3,

4 10

9. (i) E1 and E

2 occur

(ii) E1 does not occur, but E

2 occurs

(iii) Either E1 or E

2, or both E

1 and E

2 occurs

(iv) Either E1 or E

2 occurs, but not both

10. (i)1

3, (ii)

23

1812.

3

213. Rs 0.50 14.

1

10

15. Expectation = Rs 0.65 16.85

15317.

7

15

18.5

919.

1

27072520.

5

1621.

7

128

22.4547

819223.

891–

10 24. (i) .1118 (ii) .4475

25. (i)8

15, (ii)

14 1,

15 15, (iii) 1 26. 0.7 (approx.) 27. 0.18

28.1

229. X 0 1 2

P (X) .54 .42 .04

31. (i)10

49

50

(ii)8

10

45(49)

(50)(iii)

9

10

59(49)

(50)

304 MATHEMATICS

32.1

333.

9

4434.

–1–1

p

n

35. X 1 2 3 4 5 6

P(X) 36 36 36 36 36 36

36.1

2p = 37.

665

32438.

775

7776

39. not independent 41. (i)7

18, (ii)

11

1842. (i)

2

11, (ii)

9

11

43. (i) 0.49, (ii) 0.65, (iii) .314 44.7

1145.

11

21

46.1

347.

110

22148.

5

11

49. (i)1

50, (ii) 5.2, (iii) 1.7 (approx.) 50. (i) 3, (ii) 19.05

51. (i) 4.32, (ii) 61.9, (iii)15

2252. 10

53. Mean2

13= , S.D. = 0.377 54.

1

2

55. Mean = 6, Variance = 3

56. C 57. A 58. D 59. C

60. C 61. D 62. B 63. D

64. C 65. D 66. D 67. D

68. C 69. D 70. D 71. D

72. C 73. C 74. C 75. B

76. B 77. D 78. C 79. A

80. D 81. B 82. C 83. C

84. A 85. B 86. A 87. C

88. D 89. D 90. A 91. B

ANSWERS 305

92. D 93. D 94. False 95. True

96. False 97. False 98. True 99. True

100. True 101. True 102. False 103. True

104.1

3105.

10

9106.

1

10

107. ( )22 –i i i ip x p xΣ Σ 108. independent